Irreversibility analysis for flow of nanofluids with aggregation in converging and diverging channel

In the current research article, the two-dimensional, incompressible, steady fluid flow is considered. The heat transfer rate of water-based aggregated fluid between converging/diverging channels of shrinking/stretching walls due to the effects of thermal radiation has been examined. The strong static magnetic field is applied perpendicular to the radial direction. The modeled governing equations are transformed into non-linear dimensionless ordinary differential equations by considering appropriate similarity transformations. Since the obtained ODEs are strongly non-linear and the exact solution of these equations is not possible, thus we applied the numerical method RK4 combined with the shooting technique to handle the equations. The impacts of several influential parameters on velocity, temperature, and entropy generation profiles are examined graphically.

www.nature.com/scientificreports/ nonlinear ODEs. The Bruggeman model is used to obtain the thermal conductivity of aggregation. The effects of different parameters like radiation, volume fraction by nanoparticles, Reynolds number, sweep angle, stretching/ shrinking parameter, Eckert number on velocity field, temperature field and entropy generation are discussed graphically.

Formulation of governing equations
Here we considered two-dimensional steady flow between converging/diverging channels. The non-parallel walls of the channel make a sweep angle of measurement 2α at the intersection of walls. The cylindrical polar coordinate system is under consideration and the flow is measured only along the radial direction i.e., u r = [u(r, �), 0, 0] . A static magnetic field B 0 is imposed in direction at right angle to u r , Thus, the governing equations of fluid flow problem along with the appropriate boundary conditions are modeled as follows ( Fig. 1) 4,42,43 , The appropriate BCs at shrinking/stretching walls are: To convert the above system into dimensionless form, the following similarity transformations are applied.
In above Eqs. (1)(2)(3)(4)(5)(6)(7), u w is velocity at the wall of the channel, s t is stretching/shrinking parameter, α is sweep angle, u c represents the velocity of the particles along the centerline of channel, u r = (u(r, �), 0, 0) is cylindrical form of velocity component along radial direction, B is magnetic field parameter, T w represents temperature at the wall of channel, T ∞ is ambient temperature, η is angular coordinate in dimensionless form, f (η) is dimensionless velocity, θ(η) is dimensionless temperature and P is pressure. The symbols ρ a , σ a , v a , µ a , k a , ρC p a represent density, electrical conductivity, kinematic viscosity, dynamic viscosity, thermal conductivity, specific heat capacity respectively for the aggregated nanoparticles. Which is shown in Table 1. Figure 1. The schematic flow model through convergent (inflow) and divergent (outflow) channels. is (volume fraction of aggregated nanoparticles). The symbols ρ f , σ f , v f , k f , µ f , ρC p f are density, electrical conductivity, dynamic viscosity, thermal conductivity, kinematic viscosity and specific heat capacity of nanoparticles respectively and ρ s , σ s , µ s , v s , k s , ρC p s density, electrical conductivity, kinematic viscosity, dynamic viscosity, thermal conductivity and specific heat capacity of base fluid respectively. D = 1.8 is (Fractal Index), r a r p = 3.34 is (ratio of radii of aggregates to nanoparticles), φ m = 0.605 is (maximum volume fraction of nanoparticles), [η] = 2.5 is (Einstein coefficients).
The Bruggeman model was used for transforming the Maxwell model to obtain the thermal conductivity of aggregation. The aggregated thermal conductivity model was displayed as 44 : By the applications of transformations (7), Eqs. (1-5) becomes, The transformed dimensionless boundary conditions corresponding to Eqs. (5) and (6) are as follows: In Eqs. (9)(10)(11)(12) the dimensionless parameters are: χ = s t u c (non-dimensional stretching shrinking parameter), f (non-dimensional velocity),

Entropy analysis
The rate of entropy generation for the recent problem by following Bejan 45 is expressed as: Using the similarity transformations (7) and Eq. (13), the entropy generation number N g takes the form: Thermal conductivity

Solution procedure
In the current exploration the modeled physical problem is based upon conservation laws which appear in the form of PDEs. These equations are changed into the set of highly coupled nonlinear ODEs. Since the transformed equations are highly nonlinear and exact solution of these coupled equations is not available. To overcome this issue, in our work, we applied a famous numerical scheme (RKF) combined with shooting iteration technique to compute the solutions of the transformed fluid flow problem. Solutions for temperature, velocity and the entropy generation fields are observed graphically, and the impacts of several influential parameters are examined.

Results and discussions
This section is divided into three parts to study the variation in temperature, velocity, and entropy generation of fluid flow for both converging/diverging channels of shrinking/stretching walls under the effects of different parameters like sweep angle (α) , volume fraction of aggregated nanoparticles (φ a ) , stretching/shrinking parameters (χ) , Eckert number (Ec) and the Reynolds number (Re).
The impressions of pertinent parameters on velocity profile are plotted in set of Figs. 2, 3, 4, 5, 6, 7, 8, 9. Figure 2 shows that for the increasing values of opening angle α , the velocity profile decreases. Here the change is velocity is abrupt for the stretching wall near the central portion. An opposite impact of α on velocity is seen in Fig. 3 i.e., for the rise in angle α , velocity profile upsurges for the converging channel and the alteration in velocity is on the lower side for stretching wall. The similar behavior is observed for both aggregation and nonaggregation models. The Figs. 4 and 5 are plotted for the variation in velocity due to the effects of stretching/ shrinking parameter χ . It is noted that for the stretching walls the velocity profiles rise for both converging and diverging channel. This rise in velocity closer to walls is more effective as that of central portion. While for shrinking walls, the quite opposite behavior for both converging and diverging channels is observed. The variation in velocity is more prominent near walls as related to center portion as shown in Figs. 4 and 5. Moreover the velocity of non-aggregated nanoparticles is slightly higher for the case of diverging channel and a reverse impact is noticed for converging channel. The change in velocity field for increasing values of volume fraction by nanoparticles is plotted in Figs. 6 and 7. The decline in velocity is perceived for the rising values of volume  www.nature.com/scientificreports/ fraction by nanoparticles as shown in Fig. 6. The dominant rise in velocity of stretching wall is observed as likened to shrinking walls. The change in velocity for non-aggregation model is negligible as compared to aggregation model. All the effects of φ (nanoparticles volume fraction) on velocity in the situation of converging channel are quite opposite as related to divergent channel, except the seen that the change is prominent near centre as compared to walls as shown in Fig. 7. Figures 8 and 9 illustrate the impact of Reynolds number (Re) on velocity.       www.nature.com/scientificreports/ The change in the temperature field due to the impacts of different parameters like opening angle α , nanoparticles volume fraction φ a , Reynolds number Re , Eckert number Ec of both divergent and convergent channels are plotted in Figs. 10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33). The belongings of stretching and shrinking walls are also discussed for both converging and diverging channels. The fixed value of Pr is taken as 6.2. The impact of opening angle α on temperature profile is plotted in the set of Figs. 10, 11, 12, 13. Almost similar performance of temperature is observed for mounting values of α for both             www.nature.com/scientificreports/ profile declines. It is also noted that the change for diverging channel is rapid as likened to convergent channel. For all cases, the temperature profile for non-aggregated model is slightly above as compared to aggregated model. The Impact of volume fraction by nanoparticles on temperature field is designed in the set of Figs. 26, 27, 28, 29. The temperature profile increases for growing values of φ a for all the cases i.e., stretching/divergent, shrinking/          www.nature.com/scientificreports/ sweep angle α for both converging and diverging channels as shown in Figs. 34,35,36,37. Moreover, the entropy generation profile in the system for stretching walls is slightly above as compared to shrinking walls. The change in irreversibility of system is more prominent closer to the walls of the channels. The boost in entropy generation profile due to increasing values of Eckert numbers for divergent channel is observed for both stretching/shrinking walls. This alteration in the entropy generation is more prominent near          Fig. 47. From Fig. 48 it is seen that the entropy generation declines for shrinking /divergent channel. It can be observed that for the incrementing values of Reynolds number, the entropy profile increases for shrinking/convergent channel as displayed in Fig. 49. The values of entropy     Table 2 compares the outcomes of the R-K-4 method (together with shooting technique) with the results of the R-K-4 method. Both options are in perfect agreement with one another.

Conclusion
In the current exploration, the heat transfer and thermal conductivity of steady, incompressible aggregated fluid flow under the effects of magnetic force is considered. The flow is considered between two non-parallel stretching/shrinking walls. The impacts of radiation parameter (Nr), Reynolds number (Re), volume fraction   • The augmenting values of angle of elevation reduces the velocity profile for stretching/shrinking walls of diverging channel, while a reversed performance is detected for the case of converging channel of stretching/ shrinking walls. • The stretching parameter boosts the velocity profile, whereas the shrinking parameter diminishes the velocity field for converging as well as diverging channel. • The nanoparticles volume friction and Reynolds number declines the velocity profile f for stretching/shrinking walls of divergent channel and boosts the velocity profile for stretching/shrinking walls of convergent channel. • By mounting the values of sweep angle α , the temperature profile rises for all the case i.e., stretching/divergent, shrinking/divergent, stretching/convergent, and shrinking/convergent channels. • The growing values of Re and Ec upsurges the temperature field for stretching/shrinking divergent channel.
On the other hand, for stretching/shrinking convergent channel temperature profile declines by mounting values of these parameters. • The temperature profile rises for both converging and diverging channels under the effects of stretching parameter, while shrinking parameter diminishes the temperature profile for both converging and diverging channels. • For the growing values of radiation parameter and nanoparticles volume fraction, the temperature profile reduces for both converging and diverging channels of stretching/shrinking walls. • The entropy generation declines for all the values of opening angle α (either positive or negative) for the stretching/shrinking walls of channel. • The increasing values of Eckert number Ec increases the entropy profile for stretching/shrinking divergent channel and decreases for stretching/shrinking convergent channel. • There is direct variation between radiation parameter and the entropy generation profile i.e., by mounting the values of radiation parameter, irreversibility of the system increases. • Reynolds number declines the entropy profile for stretching/shrinking diverging channel and upsurges it for stretching/shrinking convergent channel.

Data availability
The data and material used and/or analysed during this study are available from the corresponding author on reasonable request.